Question

The following data gives the distribution of total monthly household expenditure of $200$ families of a village. Find the modal monthly expenditure of the families. Also, find the mean monthly expenditure.

Expenditure (in Rs.)	Frequency	Expenditure (in Rs.)	Frequency
1000-1500	$24$	3000-3500	$30$
1500-2000	$40$	3500-4000	$22$
2000-2500	$33$	4000-4500	$16$
2500-3000	$28$	4500-5000	$7$

Answer 1

Expenditure (in rupees)	$x_i$	Frequency $\left(f_i\right)$	$f_i{x}_i$
1000-1500	1250	24	30000
1500-2000	1750	40	70000
2000-2500	2250	33	74250
2500-3000	2750	28	77000
3000-3500	3250	30	97500
3500-4000	3750	22	82500
4000-4500	4250	16	68000
4500-5000	4750	7	33250
		$\Sigma f_i=200$	$\Sigma f_i{x}_i=532500$

$Mean=\frac{\Sigma f_i{x}_i}{\Sigma f_i}=\frac{532500}{200}=2662.5$

As frequency is maximum for the class 1500-2000, modal class is $1500-2000$.

$Mode=l+\frac{f_1-f_0}{2f_1-(f_0+f_2)}\times h$

where,

$l$ = lower limit of modal class =$1500$

$f_1$ = frequency of modal class = $40$

$f_0$ = frequency of class preceding the modal class = $24$

$f_2$ = frequency of class succeding the modal class = $33$

h = higher limit - lower limit =$2000-1500=500$

$Mode=1500+\frac{40-24}{2\times 40-(24+33)}\times 500$

$\Rightarrow mode=1500+\frac{16}{23}\times 500=1500+347.83=1847.83$

Hence, the mean and modal monthly expenditure of the families are $2662.5$ and $1847.83$ respectively.